Asymptotic analysis of wave problems for a cylindrical shell

PMM U.S.S.R.,Vol.44, pp.357-362 Copyright Pergamon Press Ltd.1981.Printed

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ASYMPTOTIC ANALYSIS OF WAVE PROBLEMS FOR A CYLINDRICAL SHELL* A. M. PROTSENKO

short-waves The asymptotic solutions of elasticity theory equations for long and the being propagated along the generator of a closed circular cylindrical shell, Th Cauchy problem for normal waves, are examined. Two problems are investigated: an infinite shell and the problem of edge perturbation propagation in a semi-infinite The asympA general asymptotic solution is constructed for both problems. shell. totic parameter is the shell wall thickness or the number of waves across this thickIn the first problem, the frequency spectrum is determined, and in the second, ness. Operators of the fundamental solutions are constructed for the the wave spectrum. problems considered. On the whole, the solutions for shells are close to the solution of an analogous problem for a solid cylinder /l-33/, and especially to the results of /3/ in which quadratic bundles of operators generated by the problem were investigated. Below the asymptotic solution is constructed by analogy with /4/, but the order of the approximation is higher. The results represented differ from the extensively utilized solutions based on the Timoshenko equations /5,6/. This refers expecially to the problem of edge perturbation propagation. By using asymptotic representations, the quadratic bundle of operators analogous to /3/, are reduced to quadratic bundles of matrices whose spectral properties areinvestiqatedbyusing algebraic methods and perturbation theory /7,8/. 1. Formulation of the problem. Let us introduce the notation: R is the radius 2h is the shell wall thickness, x 11ill, a and c are the velocities of the middle surface, y1 my y-1. of volumetric and shear wave propagation, and v := cz la", and A cylindrical r,O,s coordinate system is introduced, as is the corresponding displacement vector fj== (tL,iu> 20)“. Because of the cyclic periodicity the operator & is replaced by This is actually a Fourier series in 0. the symbol -in, where n is an integer. We construct the solution of the Cauchy problem in the form

O,< a~< 1 determines the degree of approximation, s (z,t, h) = act - hz is the phase of Here the wave, where hRa-Wa is the wave number, and w (h)cRa-‘Is-~ is the frequency spectrum which we will determine. The solution of the second problem on edge perturbation propagation in a semi-infinite shell (z :> 0) will be constructed in an analogous form (1.2) The expression for the wave phase s(z,1,tn) remains the same as for (1.1) but we will con01(h) struct the wave spectrum h(o), 01E (-CO,~). We shall henceforth consider the spectra ga, to depend everywhere on n as a parameter. The and 7"(w), exactly as the amplitude amplitudes fia(c, w) and regular functions of 5. 6a.(5, 1) themselves will be considered p : ,.R"-l/,-U and let the prime denote Let us introduce the dimensionless coordinate differentiation with respect to it. On the basis of (1.1) and (1.2) we can replace the operd, by the symbols icmRa-‘Ka and ators cl, and - ihRa-‘hia , after which the Lam& equations will berepresented in the rather unusual form

,

*Prikl.Matem.Mekhan.,44,No.3,507-515,198O 357

C = diag (rl, I, I),

S : tliag(1, 1, y,)

(1.3)

358

A. M. Protsenko

Let us introduce

a vector

equivalent

to the stress

vector

in the plane

r =

const (1.4)

Using

the variable

{) in place

of

r and replacing

d, by its symbol,

we obtain (1.5)

No stresses

on the free surfaces ga’

r=:R&h

p YZ X-a f

or

+ (A / p - ihlqg,

“fa

which

are written

as

I!, I’ -- x-” i_ x1-a

(1.6)

is the boundary condition for the ordinary differential equation (1.3). In conjunction with the boundary conditions (1.6), equations (1.3) form a spectral problem for a quadratic bundle of operators. On the basis of results /3/ for a simply-connected domain, we consider the spectra w(h) and h(o) to have a single condensation point at inIn turn, this permits the construction finity. of different approximations for the quadratic bundle of operators at regular points of the spectrum.

2. Long-wave approximation. Let us take CL -= 0, and let us consider I and co to be sufficiently small in absolute value. Then (1.3) is defined in a narrow domain pi (1 - x, 1 i-x) and we take the same approach as in /4/. To do this we expand t(p;) in a Taylor series around the middle surface (p -= l), while keeping up to four terms in the series. We combine conditions in the follows form: t (I+ Now

combining

x, .)

the expansions

! T (I -

%, .) = 0,

in the Taylor

7 (1, .) AL '/,X2? (1. .) _ 0 (y,‘), Let us introduce P3

and

the notation

p4 p. ~: -2x-2 (p1 ~- Np))

We differentiate p = 1

t (1 -I- x, .) -

series,

T (1 -

we obtain

z’ (1, .)

x, .)=O

two equations

1~‘i&Y

(1, .)

JJ!; m= go'"J(l..) and by using

7 0 (x”)

it and

0 (1). pa ==A;%-* (ps m;m Up,) - .4/J,1 0 (I),

(1.3) once

and then twice with

cp, -I-02p, $

respect

to

. . =: Cl, cp4 .: w”pz

(1.5) we solve

p, referring

From

(2.1) for

A1(-= .1 - iiLB

(2.2)

the result

-= 0

Components with the lowest terms are omitted in these equations. obtain equations in p1 and pz(E is the unit 3 X 3 matrix) (E - '/Go2x?c-')pz:: _-.lfp,-:./4:,,-,-0 (x2),

(2.1)

to

(2.3) (2.2) and

(E - '/,(02x*c-')p, = -Jlpo

(2.3) we

-i 0 (x')

(2.4)

Let (di(’ (< 1, which is in complete agreement with the assumption about the smallness of o and even weakens it substantially. For instance, 1 c?x' 1 can be considered a small number and then (2.4) can be solved as follows: p1 = -(E

+ Vzw2xV_?)jlilpg-1.0 (x" tm ("*ila),pz

= (E + ‘i, dx*c-‘)

(Ap, - Mp,) + 0 (x2 $mdj(“)

(2.5)

There now remains to substitute the expressions obtained into (1.3), referring it to p = 1. But first, p1 in the second expression in (2.5) should certainly be replaced by P,, by using the first expression. We finally obtain the system of linear algebraic equations IT, (1) + 012(E + "/,2TI(,) (h))lpor= 0 (112in d%“) Here

Tj (h)(j = 0, 1) are quadratic

bundles

Tj (A) = Pj - ihKj We present

PO=

just the structure

--'1(1-JJ) &7 (1-y) h(l-y) -4tP(l--y) 0 0

of the bundle

- 172

(2.6)

of matrices h2Lj,

j

= 0, 1

(2.7)

T,,(i)

-

0 0 7)(3 -

-211 - 2y) I?(3 - 41’) 1 L, : 417) 0 I

tliag(0, 1, 4 (1 - y))

Wave

T1c4) is formed

The bundle

To

problems

for a cylindrical

359

shell

as follows:

(h) :: I/,I2 (CM -

N)C-‘M

-i IV* + Al,

N = D -

ihG

The subscript (4) The matrix coefficients of the bundles To and Tlcr) are real matrices. gndicates the order of approximation of the boundary conditions (1.6). The fact is that the 0(x”) will result in an analogous result, in which order of the approximation T1(:!) (h) will to take 7"' = 0 in (2.1). An approximation replace T,(J)(A) - For this it is sufficient For such an approximation 0(x’) is used in /4/ and a simpler axisymmetric problem is solved. The matrix T1c3)is constructed thus: T1cL) will be the zero-th matrix. T,(.j, (h) = 'i,(CM -

N)C-‘.V

The bundle 7',(h) is comprised of the symmetric matrix P,, the skew-symmetric matrix and the diagonal degenerate matrix L,. K0 h the bundle T,, is Hermitian, negative-definite, and h is singular. Therefore, for real Because x2 is a The bundles T1c3) and T1(d) do not possess the property of being Hermitian. small number, perturbation theory can be used, but it would be desirable to reduce the bundle (2.6) to an analytic perturbation of the bundle T,,(h) m= T,,(h) f 02E. This can be done by linear interpolation by introducing the bundle

Let us require that such a bundle be Hermitian. This can always be done in the general case. ratio equals 0.25) the condition that T,(k) be Hermitian will be For y = '1, (the Poisson's k, E _-"I9 and k, = 1. In fact, there is an extrapolation here towards reducing the order of the approximation of the boundary conditions (1.6), which became 3kl + 4k, z= 7/3. But here the general order of approximation of the operator, 0 (x")I was conserved. Finally, discarding small terms in the right side in (2.6) and introducing the bundle perturbed quadratic bundle of T1 (A) , we arrive at the spectral problem for the analytically matrices T, (h) = T, (1) i a?[E -t x22’, (a)] (2.8) For

y ='I3

the bundle

T, is represented

2 + 36n2 + 3Gh2 --1Oi? ih I

T,(h)=&

3. initial

The

Cauchy

problem.

as follows:

Let us examine

- IOrl 18-W -2ik/

- ih 2ihu -8hZ

this problem

in the long-wave

band

when

the

data

are given. In turn, this means that the initial data are given in the form of an inverse Fourier transform in the coordinate z.:p,,(h) = m0 (--h); o @)pO (h) = @I (--h), where m0 (It) and @,,(h)are the Fourier transforms for (Pan and (Pm - For this reason the frequency spectrum h(o), hi (--oo,m)should be constructed. To,@) = I‘, (h) + db’. In addition to the bundle T,(h) in (2.8), we consider the bundle For any real h the bundle T, (h) is Hermitian and negative-definite. Therefore, there are three eigenvalues oOj"(h)> 0 to which the unitary matrix of eigenvectors Q(h) corresponds. The bundle T,(h) can be considered as an analytic perturbation of the bundle T,,(h) /7/. Then the estimate 0j2 (h) = 00,’ (1) [1 - X2ejTT, (h)ejl + 0 (Ooj4X4) (3.1) holds for the eigenvalues of this bundle Oj' (h) , where ej = ej (h) is the eigenvector of j -th column of the matrix Q(h). Too (h) or the In the limits 0(04X4) the eigenvectors of both bundles agree. We select the positive values of Oj (h), i = it 2v 3 and we form the diagnonal matrix

(3.2)

Q (h) = diag lo, (A), o2 (h), o3 @)I Now the operator

(1.1) is constructed

in the form of the diverging

From initial data in the form CD, and @I we determine solution is constructed as a convolution

'P+ and

waves

'P- and finally

q,, (R, n, 2, t) = Dn’ (z, 4 * ‘eon(4 + Dn (2, 4 * ‘PM(4 D, (z, t) =

&

1 exp (+-) Q (h)sin -a

($ Cl(h)) x

!A-’ (I.) Q*(h) da

the general

(3.4)

360

A. M. Protsenko

The dependence parameter.

on n in the last expression

under

the integral

should

be understood

as on

a

4. Problem of edge perturbations. We solve this problem, as the preceding one has been, in the low-frequency band of the perturbations applied to the edge z=Oin the p,,(o), 0)E (-co, m). To construct the form q,,(X, II!0. t) x f,,(f) with the spectral function solution, we determine the wave spectrum h(w) for which we consider the bundle T,,(h) in the following form: To (h) - p - ihI{- h2L,

P :~ I’,, -I~ or’fi T CCI’X~~‘~, 11’

I. == L, ~, "?x'L,,

I%‘,+ OI’X’K,

(4.1)

The characteristic equation is the bicubic equation 1T,(h) 1 = (1, that has six roots with possible multiplicity takenintoaccount, that are disposedsymmetrically relative tothe real and imaginary axes of the complex plane. Only three roots h, (o), j : i.2.3, located in the positive part of the real axis and opening into the lower complex half-plane, satisfy the radiation conditions. :,‘< :: matrix Let a normal Jordan form of the J (‘(11) correspond to these roots. This matrix is constructed by means of elementary divisors of the bundle T,,(k). Let us form the G Xcj matrix transformation .r6 = diag (J(o), --J (co)),and let us construct an equivalent of the bundle I',, (il). The equivalent transformation of the linear bundle *r6- h EG corresponds 6 s 6 unit matrix. to it, where E6 is the Let us introduce the auxiliary vector I in three-dimensional complex space, and let us convert the spectral problem for 7',,in the form (4.1) into a spectral problem for the linear bundle of Ii ci matrices

(4.2) For

1I, 1fO

this problem

is equivalent

to the problem

about

the spectrum

of the matrix

Here 1 r, - hE,I = - /L 1-l / T,(h) I. It hence follows that there exists a nondegenerate transformation U independent of h such that U.I,!Y 7= To /a/. Then X(o) denote a 3 X 3 matrix of eigen- and associated vectors for J(o). Let X, = diag(X (a), -X(o)) will be the matrix of eigen- and associated vectors for J, , and will be the same matrix for To. Y, = ux, the Only the upper half of the matrix Y, , the three upper rows, will correspond to to the vector vector p0 while the rest will correspond 2. Only the first three columns of 1'B will correspond to the eigenvalues Consequently, satisfying the radiation conditions. Y,, to be denoted by only the left upper 3 :,:3 block of the matrix y (0) I should be considered eigenvectors of the bundle T,,. This matrix is represented in the form Y (w) = Q (0)X (o), where Q (w) is a 3 X 3 matrix satisfying the equation Z'Q ---iKQ/ (0))- I;Qd2 (01)m= 0

(4.3)

The nondegenerate solution of this equation exists to the accuracy of the similarity transThis latter circumstance turns out to be negligible for the subsequent conformation /8/. struction. Now (1.2) can be written thus:

from the boundary The function I$, is determined written in the form of a convolution in t:

condition

qo (R, n, z, t) ~= D,* (z, t& f, (t) , I),* (z, 1) = &

‘s cxp (i +) -e_

for

z--o,

Y (0) enp [ -

and the final

i T]

Y-’ (co) do,

result

is

(4 *4)

The subscript n indicates dependence of the integrand on n as on a parameter. iR& h is replaced by An investigation of the bundle T,,0~) as an operator in which r,(h)po is a singular perturbation of the In this case the equation for is of special value. The fact is that the characteristic equation1 T, (h) I-(02Ej = 0 is equation (To (3,) + dElp, = 0. sign a biquadratic with the roots + h,, (0). &l, (0). We consider the roots with the plus The corresponding roots of the bundle satisfy the radiation conditions. T, (A) are a regular perturbation of the roots h,, and ho, and satisfy the estimate 1

hj'

(0))

-

h,]ji

(W)

1

=

0

(Wax'),

j

=

1, 2

Wave

problems

for a cylindrical

361

shell

The third root in the characteristic equation, to be denoted and is written thus in general form: of the singularity,

by

*h, (0) is a consequence

X,2 (0) = 2 (1 - y)x-Z (02 - n") Lo* - 4 (1 - y)l [UP- 4 (1 - VI b’ 7 I)1 + 0 (1) Therefore, for n> 1 and small o this root has a large imaginary part, which means the z -0. However, formation of a rapidly damped wave in a narrow boundary layer zone around for relatively large o (on the order of IZ), the formation of slow undamped waves is possible. Then a component containing z Still another singularity appears for multiple h, = h,. and the phenomenon of quasiresonance will be observed in appears in the matrix exponential, This is characterized by the fact that the amplitude maximum will be at a the wave pattern. and will subsequently drop certain distance from the site of application of the perturbation, exponentially.

5. Short-wave asymptotic. In the short-wave band h and o are sufficiently large. CL= 1 to construct the solution in the form (1.1) and (1.2), but the boundHence, we take In this case the Taylor series exary conditions (1.6) will here be given for p = x-1 f 1. pansion does not yield a satisfactory result for a small number of terms in the series. Hence, P' = p - x-1. We represent we go over to the new argument (1.3) and the boundary conditions (1.6) as follows: Cg," - i?.Gg,' - h*Sg, i 02g, = 0 (x2),

g,’ -

ihBg,

= 0 (x),

P’ =

tl

(5.1)

In such a formulation, the boundary conditions which have an error O(x) assure compliance with conditions (1.6) with an error 0(x2). Discarding the small terms in the right side of the operator (5.1), we arrive at the spectral problem for the operator describing wave propagation in an unbounded plate of thickness The solution of such a boundary value 2h. problem for an ordinary differential equation will be constructed in the form g, 7 esp (pp'). This results in such a characteristic equation 1 p2C - ihpG - b2S + w*E 1 = 0 Here

there

We construct

are six roots

in all Pl?d2 = a2 - yw2, pe,52= pJ solution in the form

the general gl=exp

q

(

1

q++exp

---
(

1

= h2 -

‘p_, L=r--R,

02

V = diag(pi, p2, PS)

(5.2)

Hence substituting (5.2) into the boundary conditions results in a homogeneous of equations Ze'cp,- Ze-Vcp_ = 0, Ze-'.cp+ - ZeVcp_ m= 0, Z = V - ihR, % == V + 3.B One of the solutions of the characteristic equation of such a system is p2 = This corresponds to a shear wave in the plane r m= const being propagated at ocity e. Two other solutions are obtained for o-+ m from the equation This is the equation for Rayleigh waves. However, we obtain the equation in 1 e[ %,-‘Z,eu

-

e-KZ,-lZ,e-u

Z,=U-iirZB,,.

Z,=lJ+ihB,,

( = 0,

U : I&l=

0

or

system

h, =

its natural IZI=IZI-0. the form

(0.

vel-

tliag(r/h" - yo2, VA" - 09) l~,I#O~

for not extremely high frequencies. This transcendental equation always has one real positive solution a,* (0) > YOZ. As regards the other solution a, (0) I it can be a complex-valued function. The latter constructions do not differ essentially from those presented Sects. 3 and 4.

6. Concluding remarks. To a known error, asymptotic solutions permit the construction of the solution for any initial data and edge perturbations. A particular case is the axisymmetric problem when n=O, considered earlier in /4/. The general operator for II- 0 is split into two in the solutions constructed. One describes normal waves, and the other torsion waves. The results agree qualitatively with the solution for a continuous cylinder /3/. Let us examine the agreement with an analogous solution on the basis of the Timoshenko equations /5,6/. We limit ourselves only to the axisymmetric case (n- 0) and we present two dispersion equations, obtained above and represented in /6/. We omit the regular perturbation of the coefficients by retaining just the singular perturbation for a higher degree of 2 (v = ':a) (02- 2"?) lhV,o2r?- A*4(5 - :!o") + 02 (30'- 8))= An analogous

equation

borrowed

from /6/ and reduced

0

to the notation

(6.1) used

above

is:

362

A. M. Protsenko

(ey- h'))hef;4%? / !I~ ).'802xz / 3 - 124(5 ~ 20")+ w? (302~~~S)] z 0

(6.2)

In both cases the first factor corresponds to the torsion problem, and the characteristic equation for the normal waves is contained in the brackets. For the Cauchy problem, the frequency spectrum w(h) agrees in practice in both equations, the difference is on the order of 0(x2). But the error in (6.1) is 0 (x" i-o"y.')with respect to the actual spectrum. There are substantial differences for the wave spectrum of the edge perturbation problem. There are two roots in (6-l), k,(w) and h, (0)), where h,(o) is the regular root and As (0) is the singular root. Let us recall that the root h,L~ 0 correspond to torsion and is identical for (6.1) and (6.2). Hence, the problem considered above for the edge perturbation is solvable for any conditions at the end z 0. In addition to h,- w in (6.2) there is the regular solution h,'-11)4e'1 x (302-~ R) / (“e -~ 5). which agrees with the analogous solution for (6.1). There is still another pseudoregular solution h,? :'/,w' ~:.0 (x2). But there is still another singular solution also h,?= ~':'(a%~)I-O(l), which agrees completely with the eighth order equation (6.2). Therefore, the wave spectrum consists of four branches, and the frequency spectrum of three, as assumed. The essential qualitative distinction here is that which cannot assure a solution in the edge perturbation problem for any edge perturbations. analysis is applicable foranydegrees In conclusion, we note that the proposed asymptotic of approximation 4. Thus for a Ilp we arrive at asymptotic solutions relative to 1 m. REFERENCES 1.

DE VAULT, G. P. and CURTIS, C. W., Elastic time-dependent end conditions, J. Acoust.

cylinder with free lateral surface Sot. Amer., Vo1.34, No.4, 1962.

2.

ZEMANEK, J., Jr., An experimental and theoretical investigation of elastic tion in a cylinder. J. Acoust. Sot. Amer., Vo1.51, No-l, Pt.2, 1972.

3.

KOSTIUCHENKO, A. G. and ORAZOV, M. G. Problems of the vibrations of an elastic semicylinder and associated self-adjoint quadratic bundles of operators. IN: Trudy, I. G. Petrovskii Seminar. Moscow Univ. Press, 1979.

4.

PROTSENKO, A. M., Wave propagation Mekhan. Tverd. Tela, No.6, 1979.

in a thin cylindrical

S. P. and WOINOWSKII-KRIEGER,S.,

Plates

shell,

and Shells,

and mixed

wave

propaga-

Izv. Akad.

Nauk

SSSR,

Fizmatgiz,

Moscow,

1963.

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TIMOSHENKO,

6.

of an infinite thin-walled cylindriTARTAKOVSKII, B. D. and MIKHAILOV, R. N., Vibrations cal shell subjected to concentrated forces, IN: Vibrations, Radiation, and Damping of Elastic Structures. "Nauka", Moscow, 1973.

7.

LANCASTER, P., Matrix Theory, "Nauka", Press, Book No. 11664, 1966).

8.

GANTMAKHER,

F., Theory

of Matrices,

Moscow,

English

1978

(See also English

translation,

Vol.1

translation,Pergamon

and 2, Chelsea,

Translated

N.Y.

1973.

by M.D.F.